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176. CMB 1998 (vol 41 pp. 225)

Vanderwerff, Jon
Mazur intersection properties for compact and weakly compact convex sets
Various authors have studied when a Banach space can be renormed so that every weakly compact convex, or less restrictively every compact convex set is an intersection of balls. We first observe that each Banach space can be renormed so that every weakly compact convex set is an intersection of balls, and then we introduce and study properties that are slightly stronger than the preceding two properties respectively.

Categories:46B03, 46B20, 46A55

177. CMB 1998 (vol 41 pp. 41)

Giner, E.
On the Clarke subdifferential of an integral functional on $L_p$, $1\leq p < \infty$
Given an integral functional defined on $L_p$, $1 \leq p <\infty$, under a growth condition we give an upper bound of the Clarke directional derivative and we obtain a nice inclusion between the Clarke subdifferential of the integral functional and the set of selections of the subdifferential of the integrand.

Keywords:Integral functional, integrand, epi-derivative
Categories:28A25, 49J52, 46E30

178. CMB 1997 (vol 40 pp. 443)

Hadwin, Don; Orhon, Mehmet
Reflective Representations and Banach C*-Modules
Suppose ${\cal A}$ is a unital $C$*-algebra and $m\colon{\cal A}\to B(X)$

Categories:47D30, 46L99

179. CMB 1997 (vol 40 pp. 488)

Maouche, Abdelaziz
Caractérisations spectrales du radical et du socle d'une paire de jordan-banach
If $f$ and $g$ are two analytic functions from a domain $D$ of the complex plane into respectively the Banach spaces $V^+$ and $V^-$, we prove that $\lambda\mapsto \Sp\bigl(f(\lambda),g(\lambda)\bigr)$ is an analytic multivalued function. From this derives the subharmonicity of the functions $\lambda\mapsto \rho_V\bigl(f(\lambda),g(\lambda)\bigr)$ and $\lambda\mapsto \log\rho_V\bigl(f(\lambda),g(\lambda)\bigr)$ where $\rho$ denotes the spectral radius. We apply these results to obtain nice caracterizations of the radical and the socle of a Banach Jordan pair, and finally we get an algebraic structural theorem.

Keywords:Spectre, rayon spectral, multifonction analytique, quasi-inverse,, paire de Jordan-Banach, radical de Jacobson, socle.
Categories:46H70, (17A15)

180. CMB 1997 (vol 40 pp. 356)

Mazet, Pierre
Principe du maximum et lemme de Schwarz, a valeurs vectorielles
Nous {\'e}tablissons un th{\'e}or{\`e}me pour les fonctions holomorphes {\`a} valeurs dans une partie convexe ferm{\'e}e. Ce th{\'e}or{\`e}me pr{\'e}cise la position des coefficients de Taylor de telles fonctions et peut {\^e}tre consid{\'e}r{\'e} comme une g{\'e}n{\'e}ralisation des in{\'e}galit{\'e}s de Cauchy. Nous montrons alors comment ce th{\'e}or{\`e}me permet de retrouver des versions connues du principe du maximum et d'obtenir de nouveaux r{\'e}sultats sur les applications holomorphes {\`a} valeurs vectorielles.

Keywords:Principe du maximum, lemme de Schwarz, points extr{émaux.
Categories:30C80, 32A30, 46G20, 52A07

181. CMB 1997 (vol 40 pp. 133)

Blackmore, T. D.
Derivations from totally ordered semigroup algebras into their duals
For a well-behaved measure $\mu$, on a locally compact totally ordered set $X$, with continuous part $\mu_c$, we make $L^p(X,\mu_c)$ into a commutative Banach bimodule over the totally ordered semigroup algebra $L^p(X,\mu)$, in such a way that the natural surjection from the algebra to the module is a bounded derivation. This gives rise to bounded derivations from $L^p(X,\mu)$ into its dual module and in particular shows that if $\mu_c$ is not identically zero then $L^p(X,\mu)$ is not weakly amenable. We show that all bounded derivations from $L^1(X,\mu)$ into its dual module arise in this way and also describe all bounded derivations from $L^p(X,\mu)$ into its dual for $1
Categories:43A20, 46M20

182. CMB 1997 (vol 40 pp. 254)

Saito, Kichi-Suke; Watatani, Yasuo
Subdiagonal algebras for subfactors II (finite dimensional case)
We show that finite dimensional subfactors do not have subdiagonal algebras unless the Jones index is one.

Categories:46K50, 46L37

183. CMB 1997 (vol 40 pp. 183)

Kepert, Andrew G.
The range of group algebra homomorphisms
A characterisation of the range of a homomorphism between two commutative group algebras is presented which implies, among other things, that this range is closed. The work relies mainly on the characterisation of such homomorphisms achieved by P.~J.~Cohen.

Categories:43A22, 22B10, 46J99

184. CMB 1997 (vol 40 pp. 129)

Badea, Catalin
Sur les caractères d'une algèbre de Banach
A new proof for the Gleason-Kahane-\.Zelazko theorem concerning the characters of a Banach algebra is given. A theorem due to P\'olya and Saxer is used instead of the Hadamard factorization theorem.

Categories:46H05, 32A15

185. CMB 1997 (vol 40 pp. 10)

Borwein, Jon; Vanderwerff, Jon
Convex functions on Banach spaces not containing $\ell_1$
There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an isomorphic copy of $\ell_1$. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions.

Categories:46A55, 46B20, 52A41
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